Suppose that $f(x)$ and $g(x)$ are functions on $[a,\infty)$, and that
$$0 \le f(x) \le g(x)$$ for all $x \in [a,\infty)$. Then
$\int_a^\infty f(x) \,dx$ will always be better behaved than
$\int_a^\infty g(x)\, dx$.
In particular:

If $\int_a^\infty g(x)\, dx$ converges, then $\int_a^\infty f(x)\, dx$ converges.
 If $\int_a^\infty f(x)\, dx$ diverges, then $\int_a^\infty g(x)\, dx$ diverges.
 If $\int_a^\infty f(x) \,dx$ converges, then we can't tell what happens to $\int_a^\infty g(x) \,dx$.
 If $\int_a^\infty g(x) \,dx$ diverges, then we can't tell what happens to $\int_a^\infty f(x) \,dx$.
